Decompositions of Rational Gabor Representations

Let $\Gamma=\langle T_{k},M_{l}:k\in\mathbb{Z}^{d},l\in B\mathbb{Z}$ be a group of unitary operators where $T_{k}$ is a translation operator and $M_{l}$ is a modulation operator acting on $L^{2}\left( \mathbb{R}^{d}\right) .$ Assuming that $B$ is a non-singular rational matrix of order $d,$ with at least one rational non-integral entry, we obtain a direct integral irreducible decomposition of the Gabor representation which is defined by the isomorphism $\pi:\left( \mathbb{Z}_{m}\times B\mathbb{Z}^{d}\right) \rtimes\mathbb{Z}^{d}\rightarrow\Gamma$ where $\pi\left( \theta,l,k\right) =e^{2\pi i\theta}M_{l}T_{k}.$ We also show that the left regular representation of \left( \mathbb{Z}_{m}\times B\mathbb{Z}% ^{d}\right) \rtimes\mathbb{Z}^{d} which is identified with $\Gamma$ via $\pi$ is unitarily equivalent to a direct sum of $\mathrm{card}\left( \left[ \Gamma,\Gamma\right] \right)$ many disjoint subrepresentations: $L_{0},L_{1},\cdots,L_{\mathrm{card}\left( \left[ \Gamma,\Gamma\right] \right) -1}.$ It is shown that for any $k\neq 1$ the subrepresentation $L_k$ of the left regular representation is disjoint from the Gabor representation. Furthermore, we prove that there is a subrepresentation $L_{1}$ of the left regular representation of $\Gamma$ which has a subrepresentation equivalent to $\pi$ if and only if $\left\vert \det B\right\vert \leq1.$ Using a central decomposition of the representation $\pi$ and a direct integral decomposition of the left regular representation, we derive some important results of Gabor theory. More precisely, a new proof for the density condition for the rational case is obtained. We also derive characteristics of vectors $f$ in $L^{2}(\mathbb{R})^{d}$ such that $\pi(\Gamma)f$ is a Parseval frame in $L^{2}(\mathbb{R})^{d}.$